In a general manner, in a wireless communication system, using electromagnetic waves as the physical transmission medium, multiple reflections (multiple paths) due to the different obstacles in the propagation space (propagation in the urban environment or in the “indoor” environment) induce phenomena of fading of the signal, i.e. that over a complete frequency range, it is possible to observe destructive interference. The relative movement of the environment with respect to the transmitter and to the receiver makes these fading phenomena stochastic. These fading channels considerably penalise the performance of conventional digital communication systems. To combat fading phenomena effectively, it is necessary that the receiver has available several replicas containing the information transmitted, replicas which have suffered mutual independent fading. The concept of diversity refers to the creation of these replicas and the order of diversity is defined as the number of “independent” replicas which the receiver has available. To mention some examples of diversity: time diversity (the information is repeated with a time difference greater than the coherence interval in order that there is decorrelation between the replicas), frequency diversity, path diversity in CDMA systems, receive antenna diversity. Since 1996, a new coding method, named Space-time coding, using several transmit and receive antennas, has allowed the use of a new form of diversity: transmit antenna diversity. Space-time coding is not the only technique resorting to several transmit and receive antennas: these techniques are combined under the name MIMO (Multiple Input Multiple Output). Recourse to Space-time coding over several antennas considerably improves the performance (owing to diversity) and information rate (owing to multiplexing) of telecommunications systems in an environment having fading (“fading channel”).
From all coherent Space-time codes (i.e. that the receiver has perfect knowledge or at least an estimate of the channel coefficients) using two transmit antennas known at the present time, the one offering the best performance is the Golden Code as described in the document: J. C. Belfiore, G. Rekaya and E. Viterbo, “The Golden Code: A 2×2 Full-Rate Space-Time Code with Non-Vanishing Determinants,” IEEE Trans. Inform. Theory (April 2005). The Golden Code is actually a maximum diversity (the transmit diversity is 2) and maximum multiplexing (2 symbols per channel use). Furthermore, it also has the highest coding gains (whatever the constellation used). In addition, it was demonstrated recently that the Golden Code can be used on an “Amplify and Forward Relay Channel” with a relay and an antenna and that it can benefit from diversity of collaboration.
The Golden Code only has excellent performance for Maximum Likelihood decoding. As for a number of space-time codes (linear codes), the Maximum Likelihood decoding of the Golden Code is reduced to the search within a region (the shaping region) of a lattice (of dimension 8 for the Golden Code), a lattice which changes according to the channel coefficients, from the point closest to the point received. In order to do this, algorithms to search for the closest point within a lattice were adapted to the decoding of Space-time codes: the Sphere Decoder and the Schnorr-Euchner enumeration. These algorithms need a highly variable number of iterations and are therefore suited only with difficulty to a hardware implementation operating in real time. Accelerations of these algorithms using pre-processing, in particular basis reductions of the LLL type, have also been studied already.
Given the complexity of decoding at Maximum Likelihood decoding, sub-optimal algorithms for decoding Space-time codes have been developed. Some of these algorithms stem from interference equalization and cancellation: this is the case with the ZF (Zero Forcing), MMSE (Minimum Mean Square Error) and ZF-DFE (Decision Feedback Equalizer) algorithms described in: J. Foschini, G. Golden, R. Valenzuela and P. Wolniansky, “Simplified processing for high spectral efficiency wireless communication employing multi-element arrays”, IEEE Journal on Selected Areas Communications, vol. 17, p 1841-1852, November 1999, the V-BLAST decoding algorithm. However, these decoders cannot benefit either from transmit diversity or receive diversity.
Recently, low complexity sub-optimal algorithms have been proposed. They use MMSE or ZF algorithms combined with an LLL basis reduction. Thanks to LLL, these algorithms can use the diversity in its entirety, which is not the case in the absence of this pre-processing. In the document: A. D. Murugan, H. El Gamal, M. O. Damen and G. Caire, “A Unified Framework for Tree Search Decoding: Rediscovering the Sequential Decoder”, IEEE Transactions on Information Theory, vol. 52, no. 5, May 2006, proved the interest in using MMSE-GDFE filtering as pre-processing in order be able to perform a decoding in the lattice without considering the “shaping” constraint and to make use reduction algorithms such as LLL. Moreover, a general formalising of searching within a tree was proposed, capable accordingly of describing all the techniques resulting from the decoding within a lattice and introducing sequential decoding techniques.
However, even this latter technique cannot optimise the advantages of the Golden Code.